That's a finite sum, and so for large enough k, the entire sum should go to 0. But if we're trying to prove that for all J, K > k, then isn't it possible to hold K fixed and take J so large that the sum doesn't actually approach 0?

Just as you said, .
That's a finite sum, and so for large enough k, the entire sum should go to 0. But if we're trying to prove that for all J, K > k, then isn't it possible to hold K fixed and take J so large that the sum doesn't actually approach 0?

But the series converges.
Therefore the sequence of partial sums converges; so it is a Cauchy sequence.
That means we done.